Pseudo algebraically closed field
In mathematics, a field is pseudo algebraically closed if it satisfies certain properties which hold for algebraically closed fields. The concept was introduced by James Ax in 1967.
Formulation
A field K is pseudo algebraically closed if one of the following equivalent conditions holds:- Each absolutely irreducible variety defined over has a -rational point.
- For each absolutely irreducible polynomial with and for each nonzero there exists such that and.
- Each absolutely irreducible polynomial has infinitely many -rational points.
- If is a finitely generated integral domain over with quotient field which is regular over, then there exist a homomorphism such that for each.
Examples
- Algebraically closed fields and separably closed fields are always PAC.
- Pseudo-finite fields and hyper-finite fields are PAC.
- A non-principal ultraproduct of distinct finite fields is PAC. Ax deduces this from the Riemann hypothesis for curves over [finite fields].
- Infinite algebraic extensions of finite fields are PAC.The PAC Nullstellensatz. The absolute Galois group of a field is profinite, hence compact, and hence equipped with a normalized Haar measure. Let be a countable Hilbertian field and let be a positive integer. Then for almost all -tuples, the fixed field of the subgroup generated by the automorphisms is PAC. Here the phrase "almost all" means "all but a set of measure zero".
- Let K be the maximal totally real Galois extension of the rational numbers and i the square root of −1. Then K is PAC.
Properties
- The Brauer group of a PAC field is trivial, as any Severi–Brauer variety has a rational point.
- The absolute Galois group of a PAC field is a projective profinite group; equivalently, it has cohomological dimension at most 1.
- A PAC field of characteristic zero is C1.